Neural networks are commonly used for classification tasks, in fact from this post it seems like that's where they shine brightest.

However, when we want to classify using neural networks, we often have the last layer to take values in $[0,1]$; typically, by taking the last layer to be the sigmoid function $x \mapsto \frac{e^x}{e^x +1}$. Is this theoretically justified? (i.e., is there an analogue to the universal approximation theorem for this case)? Why can the output of the neural network be in the range $[0,1]$ if it performs classification?

Neural networks are commonly used for classification tasks, in fact from this post it seems like that's where they shine brightest.

However, when we want to classify using neural networks, we often have the output layer to take values in $[0,1]$; typically, by taking the last layer to be the sigmoid function $x \mapsto \frac{e^x}{e^x +1}$.

Can neural networks with a sigmoid as the activation function of the output layer approximate continuous functions? Is there an analogue to the universal approximation theorem for this case?

Neural networks are commonly used for classification tasks, in fact from this post it seems like that's where they shine brightest.

When we want to classify using NNs, we just take the last layer to take values in $[0,1]$; typically, by taking the last layer to be the sigmoid function $x \mapsto \frac{e^x}{e^x +1}$. Is this theoretically justified? (i.e., is there an analogue to the universal approximation theorem for this case)?

Neural networks are commonly used for classification tasks, in fact from this post it seems like that's where they shine brightest.

However, when we want to classify using neural networks, we often have the last layer to take values in $[0,1]$; typically, by taking the last layer to be the sigmoid function $x \mapsto \frac{e^x}{e^x +1}$. Is this theoretically justified? (i.e., is there an analogue to the universal approximation theorem for this case)? Why can the output of the neural network be in the range $[0,1]$ if it performs classification?

Neural networks are commonly used for classification tasks, in fact from this post it seems like that's where they shine brightest.

When we want to classify using NNs, we just take the last layer to take values in [0,1]; typically by taking the last layer to be the sigmoid function $x \mapsto \frac{e^x}{e^x +1}$. Is this theoretically justified? (i.e., is there an analogue to the universal approximation theorem for this case)?

Ie.: Neural networks of the form $$ \operatorname{Sigmoid} \circ A \circ \sigma \circ B(x);\qquad x \in \mathbb{R}^n, $$ where $A,B$ are affine maps such that the composition $A\circ B$ is well-defined and such that $\sigma$ is an activation function and sigmoid function are applied component-wise on the vector $B(x)$.

Note: $\operatorname{Sigmoid}$ may be taken to be $\sigma$... as is commonly done.

Neural networks are commonly used for classification tasks, in fact from this post it seems like that's where they shine brightest.

When we want to classify using NNs, we just take the last layer to take values in $[0,1]$; typically, by taking the last layer to be the sigmoid function $x \mapsto \frac{e^x}{e^x +1}$. Is this theoretically justified? (i.e., is there an analogue to the universal approximation theorem for this case)?